The Canonical Form of Nonlinear Discrete-Time Models

Discrete-time models of complex nonlinear processes, whether physical, biological, or economical, are usually under the form of systems of coupled difference equations. In analyzing such systems, one of the first tasks is to find a state-space description of the process—that is, a set of state variables and the associated state equations. We present a methodology for finding a set of state variables and a canonical representation of a class of systems described by a set of recurrent discrete-time, time-invariant equations. In the field of neural networks, this is of special importance since the application of standard training algorithms requires the network to be in a canonical form. Several illustrative examples are presented.